Koszul Factorization and the Cohen-Gabber Theorem

TL;DR

This paper improves the Cohen-Gabber theorem for certain local domains by establishing a dominant, finite map with degree properties related to prime numbers, using Koszul complexes and multiplicity theory.

Contribution

It introduces a new factorization approach involving Koszul complexes to sharpen the Cohen-Gabber theorem for equicharacteristic complete local domains.

Findings

Existence of a dominant, finite map with degree prime to p

Application of Koszul complex factorization in algebraic geometry

Enhanced understanding of the structure of local domains

Abstract

We present a sharpened version of the Cohen-Gabber theorem for equicharacteristic, complete local domains (A,m,k) with algebraically closed residue field and dimension d > 0. Namely, we show that for any prime number p, Spec(A) admits a dominant, finite map to Spec(k[[X_1,...,X_d]]) with generic degree relatively prime to p. Our result follows from Gabber's original theorem, elementary Hilbert-Samuel multiplicity theory, and a "factorization" of the map induced on the Grothendieck group G_0(A) by the Koszul complex.